Valuing Exit Options

Jenna Bednar∗

University of Michiganjbednar@umich.edu

December 15, 2006

Abstract

This paper examines an important aspect of federalism: the effect of a secessionthreat on the union’s productivity. Productivity requires a compliance maintenanceregime with credible punishment. An exit option gives a government the alternativeof opting out of the union rather than suffer the disutility of a punishment. Equilibriaare characterized over a continuous range of exit option values. The results indicatethat only exit options that are superior to union membership improve utility; thoseof moderate value decrease net and individual government utility due to their harmfuleffect on compliance maintenance. A prescription that emerges from these resultsis that if the exit option is inferior to the benefit from a thriving union, membergovernments should voluntarily submit to measures that make exit as costly as possible.

In federalism we observe no perfectly harmonious unions; instead, even the most stable—the

United States since 1865, Switzerland—are characterized by near-constant quibbling and,

periodically, more serious disputes. Others, such as Canada, seem to be perennially at the

brink of rupture. An emerging body of work studies the institutional design supporting

effective federations (Filippov, Ordeshook, Shvetsova 2004; Volden 2005; de Figueiredo and

Weingast 2005; Bednar 2007). In this paper I take up a special aspect of federalism’s

problem: the effect of an exit option on the productivity of the union and the utility of the

governmental members.

∗I am grateful to audience members at the 2006 Midwest Political Science Association Meetings and the2006 American Political Science Association Meetings. I also thank, without implicating, my discussants atthese conferences: Ken Kollman, Burt Monroe, and especially John Londregan. Finally, two terrific refereesand Craig Volden provided useful comments to improve the article.

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A significant literature describes the opportunity provided by the exit option, generally

coming to the conclusion that exit options are beneficial for those holding them. They substi-

tute for voice (Hirschman 1970) by being an option to use instead of within-system protest;

without contradiction, they also increase (complement) voice (Hirschman 1993, Gelbach

2005, Clark et al 2006) by improving the threat point or bargaining position. In analyses of

decentralized systems, exit options lead to subnational gains because the subnational gov-

ernment is able to extract a greater distributional allocation from the center (Triesman 1999,

De Figueiredo and Weingast 2005). In general, the better the outside option, the better off

the government. That is, exit options improve utility.

A contrasting view builds on Schelling’s (1960) insights about the value of commitment

to make threats credible. Many solutions to collective action problems require a joint com-

mitment to punish, which becomes a collective action dilemma of its own (Axelrod 1986;

Greif, Milgrom, and Weingast 1994). Despite vows to react punitively, if called upon to

do so, an agent prefers not to punish if the punishment is costly. If an agent can exit the

system rather than punish, the commitment to punish is broken. Unless an agent can pre-

commit to follow through, a threat has no deterrent effect on others capable of foresight.

Relatedly, Elster (1979/1984) invokes the image of Ulysses binding himself to his ship’s mast

to resist the sirens’ temptations; by eliminating future options, precommitment mechanisms

can overcome the problems that are introduced by changing preferences (for example, due

to addiction or an altered state of mind). This second literature focuses on the impor-

tance of punishment for joint production. Exit, while not explicitly considered, disrupts the

punishment commitment.

In this paper I bring the cautions of the second literature to bear on the study of exit

options. If federalism is portrayed as a distributional challenge—the allocation of divisible

goods between member governments—then the application of Hirschman and related insights

is fitting. But if we consider federalism to be a problem of compliance because its productivity

(including the generation of divisible goods) is dependent upon the willingness of governments

to punish one another, then exit options have a second effect that may trump the first,

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imperiling even a mutually beneficial union. This paper will show that it is not sufficient for

the value of the union to exceed the value of exiting for a state to prefer federal membership to

independence. Or, more precisely, we need to specify the value of the union more particularly

than expected return when all member governments comply. At the moment of punishment,

a member state may choose to exit rather than incur costly punishment. Therefore the extent

of punishment possible must take exit options into account. The result is that punishments

are minimized. But with less severe punishment, high compliance cannot be maintained.

The union is less productive and utility from membership drops. A mediocre exit option—

one less beneficial than a productive union—makes the optimal productive union untenable.

Possession of a mediocre exit option results in a loss of utility.

The paper is structured in four sections. First I define federalism as a compliance prob-

lem. Second, I develop a base model of compliance, generating results about the optimal

punishment and compliance levels, and making inferences about the effect of an exit option.

Third, with a more general model we see the full effect of exit options on compliance and

utility. In the fourth section I conclude.

1 Federalism’s Compliance Problem

A political community may support federalism for a variety of reasons; broad categories

include military security, economic efficiency, or improved representation (Bednar 2007).

The assignment of authority between federal and state governments is a design problem

that depends upon the community’s specific weighting of its objectives, since some goals

require trade-offs (some demand greater centralization, some greater decentralization). While

the optimal structure of every federation will be different—each according to its unique

priorities—in each federation, the constitutional division of authority is set to achieve these

objectives. The productivity of the union—its ability to meet its goals, and therefore the

return to each member government—is a function of the degree of compliance with the

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constitutional rules allocating authority.1

While legislation itself is observable, many factors confound the ability of others to deter-

mine compliance, including the multi-dimensionality of policy space, the imperfect mapping

of specific policies to constitutional requirements, the multiplicity of legislation that con-

tributes to outcomes, and the more invisible execution of policy. We will focus on actions

where overcompliance is not possible. This assumption excludes domains such as troop requi-

sitions, where it is possible that a state could send more troops than federally requested, but

includes most domains of intergovernmental activity, such as trade barriers (where overcom-

pliance would imply privileging out-of-state firms over one’s own) or activities that generate

negative externalities, such as pollution controls, where it is not physically possible to elim-

inate pollution created by other states.

Imperfect information and costly compliance motivate opportunism. To prevent the

federation from falling apart, a variety of safeguards are available: structural, fragmenting

authority; citizen/electoral, where transgressing governments are punished by the voters; ju-

dicial, through constitutional review; and political, where an integrated party system polices

politicians within the structure of an interdependent party organization. Although these

safeguards have an advantage of targeting a particular government and so are virtually cost-

less for those not punished, none of these safeguards are singly sufficient to sustain the union,

as the punishment that each can inflict is too mild to deter major transgressions (Bednar

2007). While the above list of four safeguards may or may not be present in any particular

federation, and quite often if present do not function optimally, a fifth safeguard is present

in every federation: the capacity of member governments (federal and state) to punish one

another’s transgressions through retaliatory non-compliance, with realizations ranging from

trade barriers to civil wars. While in principle intergovernmental retaliation might be tar-

geted and its punishment effect exactly calibrated, in practice its effects are both scattered

1A unitary government does not suffer from the same compliance problem as a decentralized state, butalso cannot capture the benefits of decentralization. Any prescription of federalism must weigh the costs ofcompliance maintenance, and its feasibility, against anticipated advantages. For some states (Afghanistan isone) the attractions of federalism are dwarfed by the high cost of maintaining intergovernmental cooperationand compliance.

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and may trigger escalation. There is a reason why unions that depend exclusively on in-

tergovernmental retaliation for maintenance (such as the United States under the Articles

of Confederation) tend to underperform. Adding the mild safeguards allows a fine-tuning

of the compliance maintenance mechanisms. Nonetheless, the intergovernmental retaliation

remains necessary to ward off major constitutional breaches. Any compromise to its efficacy

will injure the union.

In this article, I show how the utility the members get from the union depends not just on

the internal efficiency of the union but also upon the participants’ outside options. Stability

and the union’s productive capacity are both affected by exit options. I show how mediocre

exit options—ones that the participants would not choose over membership in the union

under normal circumstances—detrimentally affect the productivity of the union, and as the

exit option improves, but remains below the offerings of a productive union, either no union

is possible, or it will break apart at the first bit of bad luck.

2 A Base Model of Intergovernmental Compliance

The model will treat the federal union’s performance as a function of the extent that each

member government complies with the distribution of authorities. As an example, consider

the value of a common market. Free trade between member states is mutually beneficial,

and the benefits increase with the number of states that participate. But compliance is

costly: each state would like to protect its own industries, even as it enjoys free access to

the markets of other states.

We will assume that there is a cost for the government to comply, and therefore it prefers

to shirk if it believes it will not be punished or if others will shirk as well. The benefit to each

government—its utility—is a function of the actions of other governments, its own noncom-

pliance, and any sanctions. Observations of the actions of other governments are imperfect,

reflecting the informational challenges described above; judgments of actions are based upon

an imperfect signal received by others, which may be a measure of productivity or harm.2

2In other contexts, examples of imperfect signals include the price of oil as an aggregate signal of the oil

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To induce compliance, we consider trigger mechanisms. Not meeting the target triggers

punishment. Because our interest is to explore the effect of exit on the federation’s most

severe safeguard, intergovernmental retaliation, we do not add any other constitutionally-

introduced trigger mechanisms. In general, a trigger mechanism has three components: a

threshold, which it compares the signal against, a punishment, and a duration of the pun-

ishment. In the base model, we collapse the last two elements into a single parameter, a

fine, so that the punishment is realized as a utility loss. I eliminate this simplification in the

extended model of Section 3.

To simplify the presentation, we consider a linear utility function but include curvature in

the probability of being punished.3 Actions are expressed as non-compliance, a continuous

variable x ∈ [0, 1], so that as x increases, opportunism is greater, and it is bounded on

the lower end at full compliance, keeping with our assumption that overcompliance is not

possible. With costly compliance (it involves sacrifice), each government prefers not to

comply, but benefits from the compliance of others. Specifically, here we assume that each

government’s utility is a fraction α of the sum of the contributions to the federation plus

whatever that member holds back. We assume α < 1; otherwise no collective action problem

exists. Given these assumptions, the single period utility to member 1 from its contributions

and those of others (governments 2 through n) can be written as follows:

u1(x1, x2, . . . xn) = αN∑i=1

(1− xi) + x1. (1)

Although the governments cannot monitor the levels of compliance of one another, they

can infer it, subject to uncertainty, by evaluating the signal received about one another’s

activity. To model uncertainty, we will assume that the members of the federation see a

common signal ω as an indicator of the amount of non-compliance. The value of ω is a

function of the sum of the noncompliance of the members plus a noise term ε, ω(∑xj, ε) =

production of each member of a cartel (Green & Porter 1984), while in Gilligan & Krehbiel (1987, 1989) thepolicy realization or outcome is a signal of the intent of the committee that authorized the policy, and wouldbe indexed to indicate the policy source.

3See Bednar 2006 for a full treatment of both cases and general proofs of the claims that follow.

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θ(∑xj) + ε, where θ is an increasing function. Given this formulation, the noise term could

increase or decrease the governments’ perceived noncompliance. We assume here that ε takes

some value between some minimum minε and maximum maxε according to a probability

distribution F , which has an associated density f .

If the signal exceeds a threshold, T , governments punish by engaging in retaliatory op-

portunism. It is important to keep in mind that the punishment regime is not the same as

withdrawal, or secession from the union. Players still “participate,” but they comply less in

order to punish one another with the aim of deterring non-compliance.4 To avoid confusion,

we write Pr(ω > T ) to denote the probability that ω exceeds a threshold, T . That is, pun-

ishment occurs if ω(∑xj, ε) > T . Therefore the probability of punishment equals one minus

the probability that ε is less than T −θ(∑xj), expressed formally as 1−F (T −θ(

∑xj)). In

this base model, we will assume that the utility during the punishment phase is Q, strictly

less than the utility during normal play.

The threshold T and the bounds on the error term (its minimum and maximum) de-

termine the range of the governments’ aggregate behavior,∑xj (at least in all interesting

cases). Specifically, θ(∑xj) will lie between T − maxε and T − minε. To see why, let’s

consider the alternatives. Suppose that θ(∑xj) < T −maxε. In this case, there’s no pos-

sibility of punishment: even the worst possible error term will not put the signal ω above

the threshold. Therefore the governments have an incentive to be more opportunistic. Al-

ternatively, if θ(∑xj) > T − minε, then no matter what the error term is, punishment is

certain. Here too governments might as well be more opportunistic because they maximize

immediate gain knowing that punishment is on its way anyway. Thus, in equilibrium, we

can assume that θ(∑xj) ∈ [T −maxε, T −minε].

To complete the model, we need only include a discount rate δ. We restrict attention to

stationary, symmetric equilibria in which each member makes the same contribution in each

period. This allows us to write a recursive equation that gives the value for member i (the

4Abreu, Pierce, and Stacchetti (1990) show that efficient equilibria rely on grim trigger strategies. Whiletheir result is proven for a finite action space, and so does not translate directly to this paper’s infinite actionspace, the intuition is equivalent.

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present discounted sum of utilities, denoted Vi):

Vi(x1, x2, . . . , xn) = αN∑j=1

(1− xj) + xi + δ [Pr(ω > T )(−Q) + Vi(x1, x2, . . . xn)] (2)

Solving for Vi produces:

Vi(x1, x2, . . . xn) =α

∑Nj=1(1− xj) + xi − δPr(ω > T )(Q)

1− δ(3)

Government i chooses xi to maximize this value. To solve for the first order condition, we

can exploit the fact that Pr(ω > T ) = 1 − F (T − θ(∑xj)). Taking the derivative of this

new expression gives the marginal value of opportunism.

1− α− δf(T − θ(∑xj))θ

′(∑xj)Q

1− δ(4)

Recall that f is the density function associated with the probability distribution F . If

full compliance were an equilibrium, then this expression would equal zero when∑xj = 0.

We may assume that small deviations are very difficult to notice, implying that θ′(0) ∼ 0. In

other words, at full compliance the marginal increase in the signal from a slight deviation is

approximately zero. Given that assumption, the above expression cannot equal zero at full

compliance (because α ∈ (0, 1), a basic assumption of collective action problems). Therefore

in equilibrium some noncompliance occurs.5

The intuition behind the slippage result is straightforward. A constitutional division of

powers, and therefore federalism, must be self-sustaining; it cannot rely upon outside forces

to maintain it. Any self-enforcing relationship depends upon players doing what is in their

5Models that sustain full compliance assume linear signals and uniform distributions of the error term.Given these strong assumptions, it is easy to see that full compliance is possible. The first order necessarycondition becomes:

−α + 1− δQ

1− δ.

If Q and δ are sufficiently large then this expression is negative, which implies that at the margin oppor-tunism reduces the member’s value. This aligns with the intuition from the repeated prisoners’ dilemma. Ifpunishment is sufficiently severe, no deviation need occur.

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best interest. That is, results focus on utility calculations (payoffs to players) rather than

compliance maximization. And, as we’ll see, full compliance, in general, does not provide

players with as much utility as partial compliance. The cost of maintaining full compliance,

in terms of the frequency of punishment, makes member governments prefer to tolerate a

little indiscretion rather than seek perfection.

While this is sufficient to show inherent opportunism, by making further assumptions

about the distribution of the error term and the functional form of θ which enters into the

signal, we can derive comparative statics. Our modification is to assume that the error term

is uniformly distributed—that it is equally likely to assume any value between −m and m.

This implies that f = 12m

. Recall the realistic assumption that small deviations are less likely

to be caught than larger ones. One way to capture this mathematically is to assume that

θ(x1, x2, . . . , xn) = (x1 + x2 + · · · + xn)2. Recall that ω(

∑xj, ε) = θ(

∑xj) + ε. What the

governments observe about one anothers’ behavior is a combination of signal and noise: with

this assumption that squares the sum of the noncompliance, small degrees of opportunism

produce a very small signal to noise ratio, and therefore the deviation cannot be detected.

The marginal value of opportunism can now be written as follows:

1− α− δ 12m

2(∑N

j=1 xj)Q

1− δ

Notice that the noise term, ε drops out of the derivative because its effect is independent

of the action taken by the governments. In this formulation small amounts of opportunism

pay, but the marginal value of opportunism falls as the degree of opportunism increases. In

equilibrium, the marginal value of opportunism will be exactly zero. Setting the previous

expression equal to zero gives the following symmetric equilibrium level of noncompliance:

x∗i =m(1− α)

NδQ(5)

Note that non-compliance does not go to zero, no matter how high the discount factor (as

long as it is not greater than one!). This contrasts with linear folk theorem results where

full compliance is a possibility. Likewise, even very high punishments (high Q) do not

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eliminate all opportunism. Again, as long as α < 1, opportunism is inherent. Slippage—

non-compliance—is unavoidable.

We are also now able to compute comparative statics, or the effect of altering parameters

on governmental behavior. Notice that the higher α, δ, N , and Q, the less slippage: oppor-

tunism decreases as the return on compliance increases, as patience increases, as the number

of governments increases,6 and as the penalty increases. On the other hand, opportunism

increases as the range of the noise term increases (m). These are all intuitive results.

A natural question to ask is whether this slippage is meaningful. As constructed, the

federal problem has two sources of disutility: (1) punishment periods to maintain compliance

incentives, even though no one deviated from the equilibrium, and (2) the loss due to the

equilibrium non-compliance. In the example below, the second source is a greater source of

disutility than the first.

A Numerical Example We can plug in specific numerical values for each of the param-

eters to help us see the relative effects of slippage and incentive preserving punishments.

Suppose that we have a federation of two states plus a central government (so N = 3) and

that the function θ squares the non-compliance, so that ω(x1, x2, x3, ε) = (x1 +x2 +x3)2 + ε.

We will assume that the random error term is uniformly distributed between −1 and 1 so

that m = 2. We further assume that the discount rate, δ, equals 910

, the return on collective

action α equals 710

, the punishment for triggering the threshold Q equals 2, and that the

threshold T equals 1. To solve for the equilibrium, we need only plug these values into

Equation 5 to find the equilibrium level of opportunism:

xi =(2)( 3

10)

(3)( 910

)(2).

Thus, xi = 19. Each member contributes 8

9. This creates a utility loss. If each member

contributed fully (and if we ignore punishment), each member would get utility of 2.1 ( 710

6Increasing the number of participants increases the benefits of cooperation. It is possible that it wouldalso increase monitoring costs, an effect that would encourage opportunism. The model could be amendedto make uncertainty a function of the number of government members.

10

times 3) each period. However, owing to opportunism, each member only gets 710

(89)(3) = 56

30,

a per period loss of utility equal to 730

, or approximately 0.233.

We can compare this loss from opportunism to the cost of the punishment regimes. To

determine the probability of a punishment regime, we first calculate the likelihood that the

signal ω exceeds the threshold T (set at 1). From above, we have that ω = (19+ 1

9+ 1

9)2 + ε.

This exceeds 1 if and only if ε > 89. Given our assumptions, this occurs with probability 1

18.7

If we add in our assumptions that δ = 910

and Q = 2, we get that the expected per period

loss due to incentive preserving punishment equals ( 118

)( 910

)(2), which equals 110

.

Thus, in this numerical example the utility loss due to opportunism (0.233) exceeds the

loss due to incentive preserving punishment (0.1). This need not always be the case; if the

probability of punishing or the cost of the punishment were to increase, the losses due to

opportunism would decrease and the losses dues to incentive-preserving punishment would

increase. At some point, the losses due to opportunism would become smaller than the losses

due to incentive-preserving punishment.

To evaluate the effect of an exit option it will be useful to understand the importance of

the punishment mechanism on the level of compliance, and therefore utility.

Claim 1 Given the assumptions of the model, the utility during normal play from being in

the federation increases in Q, the amount of the punishment.

PROOF: See the appendix.

Kreps (1990:517–21), who considers a numerical example with a different functional form,

finds a similar result: larger punishments lead to higher utility.

With this baseline intuition established (inherent opportunism, compliance increases inQ,

and utility increases in Q), the federalism context invites us to consider an extension missing

from more generic analyses of public goods games. In a federation, member governments

have the option of exiting the union. Assume that if a participant chooses to exit (the

7Given that m = 2, the distribution is uniform with a value of 12 . Therefore, the probability that ε lies in

an interval of length x equals x2 .

11

equivalent of not playing the game at all), its expected utility is W , a parameter that is

exogenously determined and commonly known. In this symmetric game, Wi = Wj ∀ i, j.

The decision to exit is permanent. If a player exits, assume the payoff to the remaining

participants is β ≥ 0, forever. The withdrawal payoff, W , defines a participation constraint.

We can now discuss when that constraint will bind, and the effect of having an exit option on

the overall productivity of the union. Let Qp be the amount of punishment that maximizes

utility at the start of the punishment period.

Claim 2 Given the assumptions of the model, the utility from being in the federation at the

start of the punishment regime increases up to Qp and decreases thereafter, where

Qp =

√αNψ − ψ + δψ2

1− δ

PROOF: See the appendix.

Figure 1 helps to illustrate the results to this point. The severity of punishment varies

along the x axis and the y axis is utility. Two curves are graphed: Vn represents expected

utility from the union during normal play and Vp is expected utility at the start of a punish-

ment. Claim 1 shows that utility increases asymptotically in Q, the severity of punishment,

as represented by Vn. Existing treatments of exit options consider them to be a participation

constraint, binding if the value of the exit option exceeds the value of the union. However,

this claim concerns only the value of the union in normal play—when no punishment has

been triggered. When no exit is available, it is the only value that is calculated. In equilib-

rium, all governments deviate (transgress) slightly, a minor transgression that is tolerated as

unfixable given information constraints. But from time to time, stochastic error causes the

observed behavior to cross a safeguard’s threshold, triggering the punishment regime. When

no exit is available (or when it is so low that it is not worth considering), the punishment can

be optimally severe to deter transgressions. But with the potential for exit, a government

may decide to leave the union rather than suffer the punishment. Claim 2 established that

there is an optimal finite Q∗p that maximizes utility at the start of the punishment regime.

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Q (punishment severity)

W (exit value);V (utility)

Vn

Vp

4: No union, efficient

3: No stable union

2: Suboptimal unionpossible

1: Optimal unionpossible

QmaxQp*

W 3,4

W 2,3

W 1,2

Figure 1: Punishment’s Effect on Utility

If the punishment is lessened to Q∗p, the equilibrium compliance level during normal play is

less, as seen by the plotting of Vp.

We can now compare the implications of differently valued exit options. Lines graphed

at W1,2, W2,3, and W3,4 represent cut points in exit option utility. (In the next section, I

solve for these cut points in terms of V .) These cut points define areas, labeled 1, 2, 3, and

4. Looking at Figure 1, we see that in Areas 1 and 4 the exit option has the intuitive effect:

if it is especially low, it does not affect the union or utility; if it is especially high, no union

is possible, as one would expect. But in Areas 2 and 3 its effect as a participation constraint

is more subtle, to the detriment of the union (in Area 3) and utility. We can see the effect

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duplicated in a more general model.

3 A More General Model

The mechanics of this extended model are identical to the base model except that equilibrium

strategies now define two levels of compliance: that played during normal periods, when

compliance is maximized, and that played punitively, in the punishment regime. Rather than

triggering a fine, observations that cross the threshold trigger reactionary non-compliance,

where all governments reduce their level of compliance (increase x) for a finite, known number

of periods. This modification more closely resembles intergovernmental retaliation.

Let S denote the set of equilibrium strategies. Given s ∈ S, we can define Vn(s) to be

the expected utility in normal play and Vp(s) as the expected utility at the start of the

punishment regime. Let s∗ be the strategy that maximizes Vn(s) and s be the strategy that

maximizes Vp(s). We focus on finite period punishment strategies. We can write a strategy

as s = (x, z, k) where x is the extent of non-compliance in normal play, z is the extent of

selfish play8 during a punishment regime, and k is the number of periods of punishment,

where x < z. In the following claims it will be helpful to define the function z(x, k) which

equals the shirking in each of k punishment periods that would sustain a contribution level

of x in normal play. We will assume that z(x, k) is continuous and differentiable where

defined.9 We further assume that z is defined in an open neighborhood around x∗, which is

just a technical way of saying that you could support higher levels of cooperation in normal

play, but it would not be efficient to do so. That is, slippage exists, as stated earlier.

Claim 3 Vn(s∗) > Vn(s): The contribution that maximizes expected utility at the start of

the punishment regime generates does not maximize normal play expected utility.

PROOF: Let s∗ = (x∗, z(x∗, k)). For any fixed k, once a sustainable normal period non-

compliance x is selected, it determines a unique z(x, k). It follows that z is strictly decreasing

8Since z is invoked as part of an equilibrium to induce compliance, it is normatively not accurate to callit transgression or shirking.

9If no k-period punishment regime would sustain x, then z is not defined.

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in x: as the targeted behavior admits more non-compliance, so x grows, it is easier to sustain,

so the punishment regime play, z can be relaxed, with less severe consequences. Therefore,

with a fixed k, we can define the payoff at the beginning of normal play solely as a function

of x: V kn (x) . Since x∗ maximizes V k

n (x) and since z(x, k) is defined in the neighborhood of

x∗, it follows that V kn′(x∗) = 0.

Again keeping k fixed, we can also define Vp as a function of x.

V kp (x) = (1 + δ · · ·+ δk−1)U(z(x, k)) + δkV k

n (x)

Taking the derivative with respect to x,

V kp

′(x) = (1 + δ · · ·+ δk−1)U ′(z(x, k))z′(x, k) + δkV k

n

′(x)

at x = x∗, U ′ > 0, z′ < 0, and V kn′

= 0 which implies that the derivative is negative.

Therefore, x∗ cannot maximize V kp . If Vn is concave, then x < x∗.�

Corollary 1 If Vn is concave, x < x∗.

PROOF: See proof of Claim 3.

Claim 4 Vp(s∗) < Vp(s): The strategy that maximizes normal play expected utility does not

maximize expected utility at the start of the punishment regime.

PROOF: The proof follows the same logic as that for Claim 3.

Claim 5 We can fully characterize the equilibria as a function of W . (1) For W < Vp(x∗),

contribute x∗ in normal play. Expected utility is not affected. (2) For Vp(x∗) < W < Vp(x),

contribute x, where x < x < x∗, for a utility loss. (3) For Vp(x) < W < Vn(x∗) players do

not participate in the union, for a utility loss. (4) For Vn(x∗) < W players do not participate

in the union, for a utility gain. In particular, the introduction of an exit option translates

into a net utility loss for moderate exit option utilities where Vp(x∗) < W < Vn(x

∗).

PROOF: Table 1 and Figure 2 summarize the results, indicating any change to utility from

having an exit option over the case where they do not exist.

15

Table 1: Union Viability and Utility Change for Varying Exit Option UtilitiesCase Range Contribution E.U. Utility Change Union?

1 W < Vp(x∗) x∗ Vn(x

∗) none yes2 Vp(x

∗) < W < Vp(x) x Vn(x) decreases yes3 Vp(x) < W < Vn(x

∗) n/a W decreases no4 Vn(x

∗) < W n/a W increases no

1. W < Vp(x∗): The participation constraint does not bind. The union is not affected

by the withdrawal option. Governments have a strict preference to participate in the

union. Any threat to use the withdrawal option to induce a higher payoff within the

union is not credible. Furthermore, it cannot be sustained, as players could reduce the

payoff from the punishment regime below the payoff from exit. For example, to induce

compliance, participants may wage war on one another, a mechanism more costly—and

therefore more effective—than simple non-compliance. Utility is unchanged.

2. Vp(x∗) < W < Vp(x): The constraint binds. Governments choose the equilibrium

contribution, x such that Vp(x) = W . While the union can be sustained, it is not

as productive as the union was without the exit option; the introduction of the exit

option decreases equilibrium contribution levels. In the symmetric game, all players

are strictly worse off by having the exit option.

3. Vn(x) < W < Vn(x∗): The constraint binds and makes the union impossible:10 par-

ticipants hit the logical limit of how much they can increase Vp(x) by increasing the

contribution during punishment periods. As in Case 2, the players’ utility is strictly

worse off by having the option to exit.

4. W > Vn(x∗): No union is possible, as the highest expected utility from participation

in the union is less than the players’ utility from exiting. Overall utility increases.

Corollary 2: If Vn(x) > Vp(x), then there is a discontinuous drop in utility at W = Vp(x).

10The game-theoretic prediction is for no union. However, if we allow for cognitive constraints, this regionmay explain cases of unions that break apart at the first sign of trouble.

16

1: Optimalunion

possible

2: Sub-optimalunion

possible

3: NoStableUnion

4: NoUnion;

Efficient!

Vn( ˆ x )

!

V

!

Vn(x*)

!

Vp ( ˆ x )

!

Vp (x*)

!

Vp ( ˆ x )

!

Vn(x*)

!

W

Figure 2: The Effect of an Exit Option (W) on Utility (V)

PROOF: See Figure 2. As exit option utility increases from Vp(x∗), to compete with the exit

options requires an increase in the expected utility at the start of the punishment regime,

so that m is increased. Since δzδx< 0, increasing z implies a decreased normal-play level of

compliance. This reduced compliance lowers the union’s productivity, lowering utility. At

W = Vp(x), players cannot increase their punishment play expected utility any more, and

use their exit option. In so doing, their utility drops discontinuously from Vn(x) to Vp(x).

The shaded region represents utility loss from the exit option. �

As long as the value of the exit option is greater than the expected utility at the start

of the punishment regime, the addition of the exit option affects utility and often, the

17

sustainability of the union. Without any option to quit the game, the optimal contribution

in equilibrium, while not at full compliance, is as close as possible. Contribution is sustained

by making the punishment regime severe enough that the players want to avoid it. Armed

with an option to exit the union, players first compare W to Vn(s∗), the maximal expected

utility of participation in normal periods. If the exit option exceeds this amount, naturally

they will quit the union, as Case 4 describes. However, we see that even if the value of the

exit option is less than the expected value of the union, the union might not be sustainable.

As the exit option utility just surpasses the expected utility at the start of the punishment

regime (Case 2), players might have some room to adjust the punishment regime utility

upward by lessening the severity of the punishment. Players have many options for adjusting

the expected utility at the start of the punishment period: they can reduce the severity of

the single-period punishment (by complying more), they can reduce the duration of the

punishment regime, or they can lower the threshold, thereby increasing the tolerance for

non-compliance. While any one of these modifications raises the expected utility at the start

of the punishment regime, they all will lower the maximal utility obtainable during normal

periods.11 While these adjustments may rescue the union, they come at a cost: a less severe

punishment induces less cooperative behavior from all members, reducing the utility from

the union, and therefore reducing player utility. As Case 3 describes, players are limited in

how much they can adjust the punishment utility. The presence of the exit option causes

utility to decrease and the union to dissolve. Cases 2 and 3 are worrisome: governments

lose utility as a result of having a moderately-attractive exit option. We will return to these

cases in the discussion below.

4 Discussion

In every federation exit options exist. Absence of a legal right to secession does not preclude

exit, but it may make it more costly. At times they may be so undesirable, or so costly to

11Only an increase in the level of patience could make the wait worthwhile, and in general, we assumethat the discount parameter is exogenously determined.

18

pursue, that they hardly register as “options”. At other times, they are so fantastic that

they foreclose any hope for union. But what effect does the exit option have when it is in

the grey area, a mediocre to pretty good option? This paper extends our understanding of

the influence of exit options by showing that unless the option is so desireable that it is a

“no-brainer” to leave, the presence of the exit option can only lower the utility of member

governments.

The backbone of federalism’s compliance mechanism is the credible potential for inter-

governmental retaliation. Should one member government abuse its constitutional authority,

and mild safeguards fail to deter it, other governments may retaliate with their own non-

compliance. This paper demonstrates the susceptability of intergovernmental retaliation to

exit options. Retaliatory periods are costly for all governments; metanorms are necessary to

ensure the credibility of mutual punishment. Exit options are a third alternative to punish-

ment or non-punishment, a way to avoid both the cost of punishment and the penalties of

non-punishment. The temptation to exit affects compliance in every contingency, although

governments would only quit the federation in downturns. Therefore exit options are not

universally beneficial, and may actually reduce the utility of those who possess them.

Exit options serve as a participation constraint in two ways. Existing analyses argue that

the utility from being in the union must exceed utility derived from independence. As such,

it is a participation constraint on joining the union. But when the union is self-enforcing,

sustenance (and therefore utility) depends upon the compliance enforced by credible threats

of mutual retaliation. This form of punishment is costly, and although milder mechanisms

minimize its impact, the severe punishment inflicted by intergovernmental retaliation must

exist for the union to be optimally productive. When no possibility of exit is present, optimal

punishments may be sustained. But when exit is an option, a second utility calculation is

made: the anticipated value of the union once punishment is triggered. At that moment—at

the start of a punishment regime—expected utility is lower than in normal periods, and may

dip below the value of exiting. Therefore we can define a second participation constraint:

that of sustaining the union. The punishment severity may be adjusted to boost expected

19

utility at the start of punishment above the exit value, but this compromises compliance

levels, reducing utility in normal periods. Therefore, a mediocre exit option—one that is

strictly worse than the value of the union performing optimally—may still reduce utility.

At this point, institutional tinkering (beyond self-enforcing strategies) may help to sus-

tain the union and prevent utility loss. For example, see Chen and Ordeshook (1994) for

complementary analysis of constitutional secession clauses as a means to preserve federal

unions by selecting between equilibria; the existence of a constitutional prohibition of se-

cession may affect beliefs sufficiently to induce punishment of any subunit that attempts

to secede. If collective commitment to punish secession attempts is credible, (overcoming

the same metanorm challenge that made exit attractive!), then the cost of exit is raised,

reducing the utility of it. See Figure 2: As long as W < Vn(x∗), utility is either increased

or unaffected. Elster’s insights are confirmed: Constitutional prohibitions on secession are

at the least innocuous, and under conditions specified by Chen and Ordeshook, become a

method of tying one’s hands to improve utility.

Finally, we can return to the earlier debate about exit as either substitute or complement

to voice. When voice is used as a mechanism to induce compliance, as in the federalism

case, the relationship between exit and voice is conditional. In particular, in all cases where

it is interesting to compare exit and voice—when the utility from the two are comparable—

the presence of exit reduces the efficacy of voice, or the capacity to induce cooperation.

Therefore, they are substitutes. We therefore confirm Hirschman’s (1970) original intuition:

exit reduces voice.

20

Appendix

Proof of Claim 1:

Recalling Equation 5, the equilibrium amount of opportunism is given by the following

expression.

x∗i =m(1− α)

NδQ

We will assume that T = 1 and that ε is uniformly distributed in [0, 1]. Setting T = 1

is optimal given this error structure because if T > 1, participants would have an incentive

to be more opportunistic, thus lowering the benefits of begin part of the federation. And if

T < 1, the probability of punishment would increase but there would be no effect on the

marginal probability of punishment. Therefore, we’re solving for the optimal mechanism.

Recall that the value function equals

Vi(x1, x2, . . . xn) =α

∑Nj=1(1− xj) + xi − δPr(ω > T )Q

1− δ

If we plug in our value for xi we get the following expression for the value function:

Vi(x1, x2, . . . xn) =αN(1− m(1−α)

NδQ) + m(1−α)

NδQ− δm

2(1−α)2

δ2Q2 Q

1− δ

To simplify the notation, let ψ = m(1−α)Nδ

. We can rewrite the previous expression as follows:

Vi(x1, x2, . . . xn) =αN(1− ψ

Q) + ψ

Q− δ(ψ

2

Q)

1− δ

We want to choose Q to maximize this expression. Taking the derivative with respect to Q

gives the following expression.

αNψQ2 − ψ

Q2 + δ ψ2

Q2

1− δ

Since αN > 1 by assumption, this expression is always positive. This implies that the higher

is Q, the higher is utility from the union in expectation.�

21

Proof of Claim 2:

If we look at the utility to a participant at the beginning of a punishment regime, what

we call Vp(i), we get the following expression.

Vp(i)(x1, x2, ..xn) =αN(1− ψ

Q) + ψ

Q− δ(ψ

2

Q)

1− δ−Q

If we take the derivative of this expression we get

αNψQ2 − ψ

Q2 + δ ψ2

Q2

1− δ− 1

Setting this expression equal to zero gives

αNψ − ψ + δψ2

Q2= (1− δ)

Which if we solve for the Qp that maximizes the pre-punishment regime payoff, we get

Qp =

√αNψ − ψ + δψ2

1− δ

Note that the derivative increases up to Qp and decreases thereafter. Thus Qp maximizes

Vp.�

22

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